Let us describe several testing methods for white noise. The most classic approach is to use empirical ACF. Earlier, we mentioned that empirical ACF is a consistent and asymptotically normal estimate of the true ACF. In particular, in case of the white noise, we have the following convergence in distribution, see [1]}:
\(\hat{\gamma }(k) \rightarrow 0,\quad N^{1/2}\hat{\gamma }(k) \rightarrow \mathcal {N}(0, 1),\quad N \rightarrow \infty ,\quad k = 1, 2, \ldots \) 
